NCERT Class 10 Maths Chapter 1: Real Numbers Cheatsheet, Quiz & Solutions
Formula, Solutions, Examples, Quiz
Updated: 11 months ago
NCERT Class 10 Maths Chapter 1: Real Numbers Cheatsheet
Key Concepts & Formulas
- Fundamental Theorem of Arithmetic: Every composite number can be expressed as a product of primes, and this factorisation is unique, apart from the order in which the prime factors occur.
- HCF (Highest Common Factor): Product of the smallest power of each common prime factor in the numbers.
- LCM (Least Common Multiple): Product of the greatest power of each prime factor involved in the numbers.
- Relation: \( \text{HCF}(a, b) \times \text{LCM}(a, b) = a \times b \) (for two numbers).
- Irrational Numbers Proof: For prime \( p \), \( \sqrt{p} \) is irrational. Assume \( \sqrt{p} = \frac{a}{b} \) (coprime \( a,b \)), leads to contradiction using theorem: if \( p \) divides \( a^2 \), \( p \) divides \( a \).
- Composite Numbers: Numbers that can be expressed as product of primes.
- Ending with Zero: Requires factors 2 and 5 in prime factorization.
Quick Reference Table
| Concept | Formula/Method | Example |
|---|---|---|
| Prime Factorization | Break into product of primes | \( 32760 = 2^3 \times 3^2 \times 5 \times 7 \times 13 \) |
| HCF | Smallest powers of common primes | \( \text{HCF}(6,20) = 2^1 = 2 \) |
| LCM | Greatest powers of all primes | \( \text{LCM}(6,20) = 2^2 \times 3 \times 5 = 60 \) |
| HCF for Three Numbers | Same method extended | \( \text{HCF}(6,72,120) = 2 \times 3 = 6 \) |
| LCM for Three Numbers | Greatest powers | \( \text{LCM}(6,72,120) = 2^3 \times 3^2 \times 5 = 360 \) |
| Irrationality Proof | Contradiction assuming rational | For \( \sqrt{2} \): Assume \( \frac{a}{b} \), \( 2b^2 = a^2 \), 2 divides a and b, contradiction. |
| Composite Check | Product of smaller integers >1 | \( 7 \times 11 \times 13 + 13 = 13(7 \times 11 + 1) = 13 \times 78 \), composite. |
Tips & Advice
Prime Factorization First: Always start with prime factorization for HCF, LCM, or checking divisibility.
Irrational Proofs: Use proof by contradiction and the theorem that if a prime divides a square, it divides the number.
HCF/LCM Relation: Use to find one if the other is known, e.g., \( \text{LCM} = \frac{\text{product}}{\text{HCF}} \).
Ending Digits: For ending with zero, check for 2 and 5 in factors; otherwise impossible.
Verify Solutions: After calculating HCF/LCM, verify product equals numbers' product.
Composite Expressions: Factor out common terms to show compositeness.
Real Numbers Quick Tips
- Unique Factorization: Prime factors are unique except for order.
- HCF × LCM = Product: Only for two numbers; not directly for three or more.
- Irrational Sums/Products: If rational + irrational = rational, leads to contradiction.
- Cyclic Events: Time to meet again is LCM of individual times.
- Prove Irrational: Assume \( \frac{p}{q} \), square, show common factor contradiction.
Real Numbers FAQ
-
What is the Fundamental Theorem of Arithmetic?Every composite number can be uniquely factorized into primes, apart from order.
-
How to find HCF and LCM using prime factorization?HCF: smallest powers of common primes; LCM: greatest powers of all primes.
-
Why is \( \sqrt{2} \) irrational?Proof by contradiction: assuming rational leads to 2 dividing both numerator and denominator infinitely.
-
Can a number like 6^n end with digit 0?No, lacks factor 5 in prime factorization.
-
What is the relation between HCF and LCM?For two numbers a and b, HCF(a,b) × LCM(a,b) = a × b.
HCF & LCM Calculator
Additional Informatics: Key Theorems and Applications
- Theorem 1.2: If p (prime) divides a^2, then p divides a (a positive integer).
- Applications of FTA: Proving irrationality (e.g., \( \sqrt{2}, \sqrt{3} \)), determining decimal expansions of rationals.
- Euclid's Division Algorithm: Any positive integer a = bq + r, 0 ≤ r < b; used for HCF via repeated division.
- Real Numbers Classification: Rationals (p/q form) and Irrationals (non-repeating, non-terminating decimals).
- History Note: Carl Friedrich Gauss proved the uniqueness in FTA in 1801.
Real Numbers Speed Quiz
Test your knowledge with 5 questions from NCERT Class 10 Maths Chapter 1! You have 30 seconds per question.
Solutions to Exercises
Exercise 1.1
1. Express each number as a product of its prime factors:
(i) 140 = \( 2^2 \times 5 \times 7 \)
(ii) 156 = \( 2^2 \times 3 \times 13 \)
(iii) 3825 = \( 3^2 \times 5^2 \times 17 \)
(iv) 5005 = \( 5 \times 7 \times 11 \times 13 \)
(v) 7429 = \( 17 \times 19 \times 23 \)
2. Find the LCM and HCF of the following pairs of integers and verify that LCM × HCF = product of the two numbers.
(i) 26 = \( 2 \times 13 \), 91 = \( 7 \times 13 \), HCF = 13, LCM = \( 2 \times 7 \times 13 = 182 \), Verification: \( 13 \times 182 = 2366 \), \( 26 \times 91 = 2366 \)
(ii) 510 = \( 2 \times 3 \times 5 \times 17 \), 92 = \( 2^2 \times 23 \), HCF = 2, LCM = \( 2^2 \times 3 \times 5 \times 17 \times 23 = 23460 \), Verification: \( 2 \times 23460 = 46920 \), \( 510 \times 92 = 46920 \)
(iii) 336 = \( 2^4 \times 3 \times 7 \), 54 = \( 2 \times 3^3 \), HCF = \( 2 \times 3 = 6 \), LCM = \( 2^4 \times 3^3 \times 7 = 3024 \), Verification: \( 6 \times 3024 = 18144 \), \( 336 \times 54 = 18144 \)
3. Find the LCM and HCF of the following integers by applying the prime factorisation method.
(i) 12 = \( 2^2 \times 3 \), 15 = \( 3 \times 5 \), 21 = \( 3 \times 7 \), HCF = 3, LCM = \( 2^2 \times 3 \times 5 \times 7 = 420 \)
(ii) 17 = 17, 23 = 23, 29 = 29, HCF = 1, LCM = \( 17 \times 23 \times 29 = 11339 \)
(iii) 8 = \( 2^3 \), 9 = \( 3^2 \), 25 = \( 5^2 \), HCF = 1, LCM = \( 2^3 \times 3^2 \times 5^2 = 1800 \)
4. Given that HCF (306, 657) = 9, find LCM (306, 657).
LCM = \( \frac{306 \times 657}{9} = 22338 \)
5. Check whether 6^n can end with the digit 0 for any natural number n.
No, because \( 6^n = 2^n \times 3^n \), it has no factor of 5, so not divisible by 10, hence cannot end with 0.
6. Explain why 7 × 11 × 13 + 13 and 7 × 6 × 5 × 4 × 3 × 2 × 1 + 5 are composite numbers.
7 × 11 × 13 + 13 = 13(7 × 11 + 1) = 13 × 78, product of two numbers greater than 1, so composite.
7 × 6 × 5 × 4 × 3 × 2 × 1 + 5 = 5040 + 5 = 5045 = 5 × 1009, product of two numbers greater than 1, so composite.
7. There is a circular path around a sports field. Sonia takes 18 minutes to drive one round of the field, while Ravi takes 12 minutes for the same. Suppose they both start at the same point and at the same time, and go in the same direction. After how many minutes will they meet again at the starting point?
They meet again after LCM(18,12) minutes. 18 = \( 2 \times 3^2 \), 12 = \( 2^2 \times 3 \), LCM = \( 2^2 \times 3^2 = 36 \) minutes.
Exercise 1.2
1. Prove that \( \sqrt{5} \) is irrational.
Assume \( \sqrt{5} = \frac{a}{b} \) where a and b are coprime integers, b ≠ 0. Then \( a^2 = 5b^2 \). So 5 divides a^2, hence 5 divides a (by Theorem 1.2). Let a = 5c, then \( 25c^2 = 5b^2 \), so \( b^2 = 5c^2 \), 5 divides b^2, hence b. Contradiction as a and b coprime. Thus \( \sqrt{5} \) irrational.
2. Prove that \( 3 + 2\sqrt{5} \) is irrational.
Assume \( 3 + 2\sqrt{5} = r \) rational. Then \( 2\sqrt{5} = r - 3 \) rational, so \( \sqrt{5} = \frac{r-3}{2} \) rational, contradiction since \( \sqrt{5} \) irrational.
3. Prove that the following are irrationals:
(i) \( \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2} \). Assume rational, then \( \sqrt{2} \) = rational × 2 = rational, contradiction.
(ii) \( 7\sqrt{5} \). Assume rational, then \( \sqrt{5} \) = rational / 7 = rational, contradiction.
(iii) \( 6 + \sqrt{2} \). Assume rational, then \( \sqrt{2} \) = rational - 6 = rational, contradiction.
As an Amazon Associate, ProSyllabus earns from qualifying purchases. Prices shown are subject to change.
Test your CBSE Class 10 Board Examination prep
Dual AI-verified questions Real exam pattern First quiz free
Class 10 Science — Chemical Reactions and Equations (Practice Quiz)
Class 10 English — Nelson Mandela: Long Walk to Freedom (Practice Quiz)
Class 10 English — A Triumph of Surgery (Practice Quiz)
Class 10 English — A Letter to God (Practice Quiz)
Class 10 Maths — Real Numbers (Practice Quiz)
Chemical Reactions and Equations Fundamentals — Free CBSE Class 10 Board Examination Quiz
कक्षा 10 हिंदी — सूरदास के पद (अभ्यास प्रश्नोत्तरी)
Class 10 Social Science — Power-sharing (Practice Quiz)
Class 10 Social Science — Resources and Development (Practice Quiz)
कक्षा 10 संस्कृत — शुचिपर्यावरणम् (अभ्यास प्रश्नोत्तरी)
कक्षा 10 हिंदी — नेताजी का चश्मा (अभ्यास प्रश्नोत्तरी)
Class 10 English — Dust of Snow (Practice Quiz)
Class 10 Social Science — Sectors of the Indian Economy (Practice Quiz)
Class 10 Social Science — Federalism (Practice Quiz)
Class 10 Social Science — Minerals and Energy Resources (Practice Quiz)
Class 10 Social Science — Agriculture (Practice Quiz)
Class 10 Social Science — Water Resources (Practice Quiz)
Class 10 Science — Electricity (Practice Quiz)
Class 10 Science — How do Organisms Reproduce? (Practice Quiz)
Class 10 Maths — Coordinate Geometry (Practice Quiz)
Class 10 Maths — Arithmetic Progressions (Practice Quiz)
Class 10 Maths — Quadratic Equations (Practice Quiz)
Class 10 Maths — Pair of Linear Equations in Two Variables (Practice Quiz)
Reading Comprehension Passages Fundamentals | CBSE Class 10 Board Examination
कक्षा 10 संस्कृत — रचना-प्रयोगः (अभ्यास प्रश्नोत्तरी)
कक्षा 10 संस्कृत — वाच्य-परिवर्तनम् (अभ्यास प्रश्नोत्तरी)
कक्षा 10 संस्कृत — कारक एवं विभक्ति (अभ्यास प्रश्नोत्तरी)
कक्षा 10 संस्कृत — समास परिचय (अभ्यास प्रश्नोत्तरी)
कक्षा 10 संस्कृत — प्रत्ययाः (अभ्यास प्रश्नोत्तरी)
कक्षा 10 संस्कृत — अव्ययानि (अभ्यास प्रश्नोत्तरी)
कक्षा 10 संस्कृत — उपसर्गाः (अभ्यास प्रश्नोत्तरी)
कक्षा 10 संस्कृत — धातुरूप परिचय (अभ्यास प्रश्नोत्तरी)
कक्षा 10 संस्कृत — शब्दरूप परिचय (अभ्यास प्रश्नोत्तरी)
कक्षा 10 संस्कृत — सन्धिः (अभ्यास प्रश्नोत्तरी)
कक्षा 10 संस्कृत — संज्ञा एवं परिभाषा (अभ्यास प्रश्नोत्तरी)
कक्षा 10 संस्कृत — वर्णविचारः (अभ्यास प्रश्नोत्तरी)
कक्षा 10 संस्कृत — भूकंपविभीषिका (अभ्यास प्रश्नोत्तरी)
कक्षा 10 संस्कृत — सूक्तयः (अभ्यास प्रश्नोत्तरी)
कक्षा 10 संस्कृत — विचित्रः साक्षी (अभ्यास प्रश्नोत्तरी)
कक्षा 10 संस्कृत — सौहार्दं प्रकृतेः शोभा (अभ्यास प्रश्नोत्तरी)
कक्षा 10 संस्कृत — सुभाषितानि (अभ्यास प्रश्नोत्तरी)
कक्षा 10 संस्कृत — जननी तुल्यवत्सला (अभ्यास प्रश्नोत्तरी)
कक्षा 10 संस्कृत — शिशुलालनम् (अभ्यास प्रश्नोत्तरी)
कक्षा 10 संस्कृत — व्यायामः सर्वदा पथ्यः (अभ्यास प्रश्नोत्तरी)
कक्षा 10 संस्कृत — बुद्धिर्बलवती सदा (अभ्यास प्रश्नोत्तरी)
कक्षा 10 हिंदी — टोपी शुक्ला (अभ्यास प्रश्नोत्तरी)
कक्षा 10 हिंदी — सपनों के-से दिन (अभ्यास प्रश्नोत्तरी)
कक्षा 10 हिंदी — हरिहर काका (अभ्यास प्रश्नोत्तरी)
कक्षा 10 हिंदी — पतझर में टूटी पत्तियाँ, कारतूस (अभ्यास प्रश्नोत्तरी)
कक्षा 10 हिंदी — अब कहाँ दूसरे के दुख से दुखी होने वाले (अभ्यास प्रश्नोत्तरी)
कक्षा 10 हिंदी — गिरगिट (अभ्यास प्रश्नोत्तरी)
कक्षा 10 हिंदी — तीसरी कसम के शिल्पकार शैलेंद्र (अभ्यास प्रश्नोत्तरी)
कक्षा 10 हिंदी — तताँरा-वामीरो कथा (अभ्यास प्रश्नोत्तरी)
कक्षा 10 हिंदी — डायरी का एक पन्ना (अभ्यास प्रश्नोत्तरी)
कक्षा 10 हिंदी — बड़े भाई साहब (अभ्यास प्रश्नोत्तरी)
कक्षा 10 हिंदी — तोप, कर चले हम फ़िदा, आत्मत्राण (अभ्यास प्रश्नोत्तरी)
कक्षा 10 हिंदी — मधुर-मधुर मेरे दीपक जल (अभ्यास प्रश्नोत्तरी)
कक्षा 10 हिंदी — पर्वत प्रदेश में पावस (अभ्यास प्रश्नोत्तरी)
कक्षा 10 हिंदी — मनुष्यता (अभ्यास प्रश्नोत्तरी)
कक्षा 10 हिंदी — बिहारी के दोहे (अभ्यास प्रश्नोत्तरी)
कक्षा 10 हिंदी — मीरा के पद (अभ्यास प्रश्नोत्तरी)
कक्षा 10 हिंदी — कबीर की साखी (अभ्यास प्रश्नोत्तरी)
कक्षा 10 हिंदी — मैं क्यों लिखता हूँ (अभ्यास प्रश्नोत्तरी)
कक्षा 10 हिंदी — साना-साना हाथ जोड़ि (अभ्यास प्रश्नोत्तरी)
कक्षा 10 हिंदी — माता का अँचल (अभ्यास प्रश्नोत्तरी)
कक्षा 10 हिंदी — संस्कृति (अभ्यास प्रश्नोत्तरी)
कक्षा 10 हिंदी — नौबतखाने में इबादत (अभ्यास प्रश्नोत्तरी)
कक्षा 10 हिंदी — एक कहानी यह भी (अभ्यास प्रश्नोत्तरी)
कक्षा 10 हिंदी — लखनवी अंदाज़ (अभ्यास प्रश्नोत्तरी)
कक्षा 10 हिंदी — बालगोबिन भगत (अभ्यास प्रश्नोत्तरी)
कक्षा 10 हिंदी — संगतकार (अभ्यास प्रश्नोत्तरी)
कक्षा 10 हिंदी — यह दंतुरित मुसकान, फसल (अभ्यास प्रश्नोत्तरी)
कक्षा 10 हिंदी — उत्साह, अट नहीं रही है (अभ्यास प्रश्नोत्तरी)
कक्षा 10 हिंदी — आत्मकथ्य (अभ्यास प्रश्नोत्तरी)
कक्षा 10 हिंदी — राम-लक्ष्मण-परशुराम संवाद (अभ्यास प्रश्नोत्तरी)
Class 10 English — For Anne Gregory (Practice Quiz)
Class 10 English — The Tale of Custard the Dragon (Practice Quiz)
Class 10 English — Fog (Practice Quiz)
Class 10 English — The Trees (Practice Quiz)
Class 10 English — Amanda! (Practice Quiz)
Class 10 English — The Ball Poem (Practice Quiz)
Class 10 English — How to Tell Wild Animals (Practice Quiz)
Class 10 English — A Tiger in the Zoo (Practice Quiz)
Class 10 English — Fire and Ice (Practice Quiz)
Class 10 Health & PE — Doping and Ethics in Sports (Practice Quiz)
Class 10 Health & PE — Adolescent Health and Wellbeing (Practice Quiz)
Class 10 Health & PE — Diseases: Communicable and Non-communicable (Practice Quiz)
Class 10 Health & PE — The Olympic Movement (Practice Quiz)
Class 10 Health & PE — Common Sports Injuries and Prevention (Practice Quiz)
Class 10 Health & PE — First Aid and Safety Education (Practice Quiz)
Class 10 Health & PE — Yoga and Healthy Living (Practice Quiz)
Class 10 Health & PE — Posture and Postural Deformities (Practice Quiz)
Class 10 Health & PE — Food and Nutrition for Health (Practice Quiz)
Class 10 Health & PE — Effects of Exercise on the Body Systems (Practice Quiz)
Class 10 Health & PE — Physical Fitness, Wellness and Lifestyle (Practice Quiz)
Class 10 Health & PE — Physical Education: Meaning and Importance (Practice Quiz)
Class 10 English — The Book That Saved the Earth (Practice Quiz)
Class 10 English — Bholi (Practice Quiz)
Class 10 English — The Necklace (Practice Quiz)
Class 10 English — The Making of a Scientist (Practice Quiz)
Class 10 English — Footprints Without Feet (Practice Quiz)
Class 10 English — A Question of Trust (Practice Quiz)
Class 10 English — The Midnight Visitor (Practice Quiz)
Class 10 English — The Thief’s Story (Practice Quiz)
Class 10 English — The Proposal (Practice Quiz)
Class 10 English — The Sermon at Benares (Practice Quiz)
Class 10 English — Madam Rides the Bus (Practice Quiz)
Class 10 English — Mijbil the Otter (Practice Quiz)
Class 10 English — Glimpses of India (Practice Quiz)
Class 10 English — From the Diary of Anne Frank (Practice Quiz)
Class 10 English — Two Stories about Flying (Practice Quiz)
Class 10 Social Science — Consumer Rights (Practice Quiz)
Class 10 Social Science — Globalisation and the Indian Economy (Practice Quiz)
Class 10 Social Science — Money and Credit (Practice Quiz)
Class 10 Social Science — Development (Practice Quiz)
Class 10 Social Science — Outcomes of Democracy (Practice Quiz)
Class 10 Social Science — Political Parties (Practice Quiz)
Class 10 Social Science — Gender, Religion and Caste (Practice Quiz)
Class 10 Social Science — Lifelines of National Economy (Practice Quiz)
Class 10 Social Science — Manufacturing Industries (Practice Quiz)
Class 10 Social Science — Forest and Wildlife Resources (Practice Quiz)
Class 10 Social Science — Print Culture and the Modern World (Practice Quiz)
Class 10 Social Science — The Age of Industrialisation (Practice Quiz)
Class 10 Social Science — The Making of a Global World (Practice Quiz)
Class 10 Social Science — Nationalism in India (Practice Quiz)
Class 10 Social Science — The Rise of Nationalism in Europe (Practice Quiz)
Class 10 Science — Our Environment (Practice Quiz)
Class 10 Science — Magnetic Effects of Electric Current (Practice Quiz)
Class 10 Science — The Human Eye and the Colourful World (Practice Quiz)
Class 10 Science — Light – Reflection and Refraction (Practice Quiz)
Class 10 Science — Heredity (Practice Quiz)
Class 10 Science — Control and Coordination (Practice Quiz)
Class 10 Science — Life Processes (Practice Quiz)
Class 10 Science — Carbon and its Compounds (Practice Quiz)
Class 10 Science — Metals and Non-metals (Practice Quiz)
Class 10 Science — Acids, Bases and Salts (Practice Quiz)
Class 10 Maths — Probability (Practice Quiz)
Class 10 Maths — Statistics (Practice Quiz)
Class 10 Maths — Surface Areas and Volumes (Practice Quiz)
Class 10 Maths — Areas Related to Circles (Practice Quiz)
Class 10 Maths — Circles (Practice Quiz)
Class 10 Maths — Some Applications of Trigonometry (Practice Quiz)
Class 10 Maths — Introduction to Trigonometry (Practice Quiz)
Class 10 Maths — Triangles (Practice Quiz)
Class 10 Maths — Polynomials (Practice Quiz)
Political Science: Democratic Politics II Practice Quiz | CBSE Class 10 Board Examination
Economics: Understanding Economic Development Practice Quiz | CBSE Class 10 Board Examination
Hindi A/B (Kshitij II / Sparsh II / Kritika II / Sanchayan II) Practice Quiz | CBSE Class 10 Board Examination
Carbon and Its Compounds Advanced Challenge | CBSE Class 10 Board Examination
Group Discussions
No forum posts available.
